COURSE UNIT TITLE

: ADVANCED TOPOLOGY AND GEOMETRY II

DEGREE PROGRAMMES

Description of Individual Course Units

Course Unit Code Course Unit Title Type Of Course D U L ECTS
MAT 6064 ADVANCED TOPOLOGY AND GEOMETRY II ELECTIVE 3 0 0 8

Offered By

Graduate School of Natural and Applied Sciences

Level of Course Unit

Second Cycle Programmes (Master's Degree)

Course Coordinator

ASSISTANT PROFESSOR MURAT ALTUNBULAK

Offered to

Mathematics (English)
Mathematics (English)

Course Objective

The aim of this course is to give an interest of topological invariants of a space in algebraic topology with aid of homology theory. We intend to understand the geometric concepts by using algebraic tools.

Learning Outcomes of the Course Unit

1   will be able to know notions of singular homology, cohomology and homotopy groups
2   will be able to calculate the homology groups of naturally occurring topological spaces
3   will be able to use homology groups to say something about the homotopy type of a topological space
4   will be able to know duality theorems
5   will be able to know the relations between homotopy and homology groups

Mode of Delivery

Face -to- Face

Prerequisites and Co-requisites

None

Recomended Optional Programme Components

None

Course Contents

Week Subject Description
1 Standard simplex, an affine sigular simplex, The singular homology group
2 CW complexes
3 Singular homology, Subdivision
4 The Mayer-Vietoris sequence, Simplicial complexes, Simplicial maps
5 Multi Linear algebra, Differential forms, Integration of forms, Stokes theorem
6 Relationship to Singular homology, The de Rham s theorem
7 The cross product and the Künneth s theorem, A sign convention, A cup and cap products, The orientation bundle
8 Duality theorems
9 Duality and compact manifolds with boundary
10 Homotopy theory, Cofibrations
11 The compact-open topology, Homotopy groups
12 Fiber spaces
13 Free homotopy, Classical groups and Associated manifolds
14 The Hurzewicz theorem, The Whitehead theorem

Recomended or Required Reading

Textbook: Glen E. Bredon, Topology and Geometry, Springer, 1993, ISBN-13: 978-0387979267

References:
1. Saunders MacLane, Homology, Springer, 1995, ISBN-13: 978-3540586623
2. William S. Massey, Algebraic Topology: an introduction, Springer, 1977, ISBN-13: 978-0387902715
3. Marvin J. Greenberg and John R. Harper, Westview Press, 1981, ISBN-13: 978-0805335576
4. Edwin Henry Spanier, Algebraic Topology, Springer, 1994, ISBN-13: 978-0387944265
5. John Milnor and James D. Stasheff, Characteristic Classes, Princeton University Press, 1974, ISBN-13: 978-0691081229

Planned Learning Activities and Teaching Methods

Lecture notes, Presentation, Problem solving

Assessment Methods

SORTING NUMBER SHORT CODE LONG CODE FORMULA
1 ASG ASSIGNMENT
2 MTE MIDTERM EXAM
3 FIN FINAL EXAM
4 FCG FINAL COURSE GRADE ASG * 0.30 + MTE * 0.30 + FIN * 0.40
5 RST RESIT
6 FCGR FINAL COURSE GRADE (RESIT) ASG * 0.30 + MTE * 0.30 + RST * 0.40


Further Notes About Assessment Methods

None

Assessment Criteria

To be announced.

Language of Instruction

English

Course Policies and Rules

Attending at least 70 percent of lectures is mandatory.

Contact Details for the Lecturer(s)

Office: B-220 (Math. Dept.)
Phone: (30)1 85 92

Office Hours

To be announced.

Work Placement(s)

None

Workload Calculation

Activities Number Time (hours) Total Work Load (hours)
Lectures 14 3 42
Preparations before/after weekly lectures 14 4 56
Preparation for midterm exam 1 20 20
Preparation for final exam 1 30 30
Preparing assignments 10 5 50
Midterm 1 3 3
Final 1 3 3
TOTAL WORKLOAD (hours) 204

Contribution of Learning Outcomes to Programme Outcomes

PO/LOPO.1PO.2PO.3PO.4PO.5PO.6PO.7PO.8PO.9PO.10PO.11
LO.1433435343
LO.2434434333
LO.3444435343
LO.4444435343
LO.5444435343

CONTACT INFORMATION
Dokuz Eylül Üniversitesi Cumhuriyet Bulvarı No: 144 35210 Alsancak / İZMİR
Phone: +90(232) 412 12 12 - Fax: +90 (232) 464 81 35

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